| L(s) = 1 | − 3-s + 5-s − 2·9-s + 3·11-s − 13-s − 15-s + 3·17-s − 2·19-s + 6·23-s + 25-s + 5·27-s + 9·29-s + 8·31-s − 3·33-s + 10·37-s + 39-s + 2·43-s − 2·45-s − 3·47-s − 3·51-s + 3·55-s + 2·57-s − 12·59-s + 8·61-s − 65-s + 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.904·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s + 1.43·31-s − 0.522·33-s + 1.64·37-s + 0.160·39-s + 0.304·43-s − 0.298·45-s − 0.437·47-s − 0.420·51-s + 0.404·55-s + 0.264·57-s − 1.56·59-s + 1.02·61-s − 0.124·65-s + 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155678198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155678198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.16827472457531, −15.41583048519115, −14.73908860561527, −14.40449799829849, −13.83095254236839, −13.24446162947597, −12.44427834039593, −12.13887722211752, −11.46200529102784, −11.04242209394431, −10.33190321462186, −9.788268973469264, −9.209561840367904, −8.511537292713342, −8.061621443104979, −7.112405022007605, −6.450768950151215, −6.159730002591052, −5.365252379630362, −4.748097842219908, −4.140073605297645, −2.967835558332287, −2.676040266061490, −1.357852820988451, −0.7288560450425921,
0.7288560450425921, 1.357852820988451, 2.676040266061490, 2.967835558332287, 4.140073605297645, 4.748097842219908, 5.365252379630362, 6.159730002591052, 6.450768950151215, 7.112405022007605, 8.061621443104979, 8.511537292713342, 9.209561840367904, 9.788268973469264, 10.33190321462186, 11.04242209394431, 11.46200529102784, 12.13887722211752, 12.44427834039593, 13.24446162947597, 13.83095254236839, 14.40449799829849, 14.73908860561527, 15.41583048519115, 16.16827472457531
